How to Find a Limit in Calculus

Calculus is a mathematical discipline that is based on limits. The first lessons in any introduction to calculus course concerns limits, which is the value of a function as the index approaches a certain value. Solving for a limit can seem a daunting task at first, although it is often as simple as algebraic substitution. Some limits do not exist, and others require factoring to come to an answer, as straight substitution would lead to an undefined answer.

Write the equation out in the form lim(x->x) f(x). F(x) is the function you will be using to find the limit and x is the value at which you will find the limit. Say f(x) = (x - 2)/(x²-4).

Substitute the value X0, the value at which you are trying to find the limit, into the function f(x). For example, say you are trying to find the limit as x approaches 2. Substituting 2 into the example equation yields 0 in the denominator, or the part of the equation on the bottom, making the answer undefined.

Factor out the terms in the equation to cancel out any terms. The denominator, (x² - 4), can be factored into the terms (x + 2)(x - 2).

Substitute the factored denominator into the equation. In the example, this yields f(x) = (x - 2)/(x -2)(x + 2).

Cancel out like terms. In the example, (x - 2) can be canceled out since it appears in both the numerator and the denominator. This reduces the equation to f(x) = 1/(x + 2).

Substitute the X0 value you are trying to find the limit for into the reduced equation. In the example, the limit as x approaches 2 of f(x) = 1/(x+2) = 1/(2 + 2) = 1/4.

Tips & Warnings

Some functions cannot be factored so that the denominator reaches a value other than 0, in which case the limit does not exist.